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A triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c0229601.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c0229602.png" /> are topological spaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c0229603.png" /> is an imbedding with the following property, known as the homotopy extension property with respect to polyhedra: For any polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c0229604.png" />, any mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c0229605.png" /> and any homotopy
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c0229606.png" /></td> </tr></table>
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A triple  $  ( X, i, Y) $,
 +
where  $  X, Y $
 +
are topological spaces and  $  i: X \rightarrow Y $
 +
is an imbedding with the following property, known as the homotopy extension property with respect to polyhedra: For any polyhedron  $  K $,
 +
any mapping  $  f: Y \rightarrow K $
 +
and any homotopy
 +
 
 +
$$
 +
F:  X \times [ 0, 1]  \rightarrow  K
 +
$$
  
 
with
 
with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c0229607.png" /></td> </tr></table>
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$$
 +
F\mid  _ {X \times \{ 0 \} }  = f \circ i
 +
$$
  
 
there exists a homotopy
 
there exists a homotopy
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c0229608.png" /></td> </tr></table>
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$$
 +
G: Y \times [ 0, 1]  \rightarrow  K
 +
$$
  
 
such that
 
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c0229609.png" /></td> </tr></table>
+
$$
 +
G\mid  _ {Y \times \{ 0 \} }  = f \ \
 +
\textrm{ and } \ \
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G \circ ( i \times  \mathop{\rm id} )  = F,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c02296010.png" /></td> </tr></table>
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$$
 +
( i \times  \mathop{\rm id} ): X \times [ 0, 1]  \rightarrow  Y \times [ 0, 1].
 +
$$
  
If this property holds with respect to any topological space, then the cofibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c02296011.png" /> is known as a Borsuk pair (in fact, the term  "cofibration"  is sometimes also used in the sense of  "Borsuk pair" ). The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c02296012.png" /> is called the cofibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c02296013.png" />. The [[Mapping cylinder|mapping cylinder]] construction converts any continuous mapping into a cofibration and makes it possible to construct a sequence
+
If this property holds with respect to any topological space, then the cofibration $  ( X, i, Y) $
 +
is known as a Borsuk pair (in fact, the term  "cofibration"  is sometimes also used in the sense of  "Borsuk pair" ). The space $  Y/i ( X) $
 +
is called the cofibre of $  ( X, i, Y) $.  
 +
The [[Mapping cylinder|mapping cylinder]] construction converts any continuous mapping into a cofibration and makes it possible to construct a sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c02296014.png" /></td> </tr></table>
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$$
 +
X  \rightarrow  Y  \rightarrow  Y/i ( X)  \rightarrow  C _ {1}  \rightarrow  C _ {2}  \rightarrow \dots
 +
$$
  
of topological spaces in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c02296015.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c02296016.png" /> is the [[Suspension|suspension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c02296017.png" />) is the cofibre of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c02296018.png" /> — being converted into a cofibration, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c02296019.png" /> is the cofibre of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c02296020.png" />, etc. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c02296021.png" /> is a cofibration of pointed spaces, then for any pointed polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c02296022.png" /> the induced sequence
+
of topological spaces in which $  C _ {1} \sim SX $(
 +
$  SX $
 +
is the [[Suspension|suspension]] of $  X $)  
 +
is the cofibre of the mapping $  Y \rightarrow Y/i ( X) $—  
 +
being converted into a cofibration, $  C _ {2} \sim SY $
 +
is the cofibre of the mapping $  Y/i ( X) \rightarrow C _ {1} $,  
 +
etc. If $  ( X, i, Y) $
 +
is a cofibration of pointed spaces, then for any pointed polyhedron $  K $
 +
the induced sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022960/c02296023.png" /></td> </tr></table>
+
$$
 +
[ X, K]  \leftarrow  [ Y, K]  \leftarrow  [ Y/i ( X), K]  \leftarrow  [ C _ {1} , K]  \leftarrow \dots
 +
$$
  
 
is an exact sequence of pointed sets; all terms of this sequence, from the fourth onward, are groups, and from the seventh onward — Abelian groups.
 
is an exact sequence of pointed sets; all terms of this sequence, from the fourth onward, are groups, and from the seventh onward — Abelian groups.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
In Western literature a cofibration always means what is here called a Borsuk pair.
 
In Western literature a cofibration always means what is here called a Borsuk pair.

Latest revision as of 17:45, 4 June 2020


A triple $ ( X, i, Y) $, where $ X, Y $ are topological spaces and $ i: X \rightarrow Y $ is an imbedding with the following property, known as the homotopy extension property with respect to polyhedra: For any polyhedron $ K $, any mapping $ f: Y \rightarrow K $ and any homotopy

$$ F: X \times [ 0, 1] \rightarrow K $$

with

$$ F\mid _ {X \times \{ 0 \} } = f \circ i $$

there exists a homotopy

$$ G: Y \times [ 0, 1] \rightarrow K $$

such that

$$ G\mid _ {Y \times \{ 0 \} } = f \ \ \textrm{ and } \ \ G \circ ( i \times \mathop{\rm id} ) = F, $$

where

$$ ( i \times \mathop{\rm id} ): X \times [ 0, 1] \rightarrow Y \times [ 0, 1]. $$

If this property holds with respect to any topological space, then the cofibration $ ( X, i, Y) $ is known as a Borsuk pair (in fact, the term "cofibration" is sometimes also used in the sense of "Borsuk pair" ). The space $ Y/i ( X) $ is called the cofibre of $ ( X, i, Y) $. The mapping cylinder construction converts any continuous mapping into a cofibration and makes it possible to construct a sequence

$$ X \rightarrow Y \rightarrow Y/i ( X) \rightarrow C _ {1} \rightarrow C _ {2} \rightarrow \dots $$

of topological spaces in which $ C _ {1} \sim SX $( $ SX $ is the suspension of $ X $) is the cofibre of the mapping $ Y \rightarrow Y/i ( X) $— being converted into a cofibration, $ C _ {2} \sim SY $ is the cofibre of the mapping $ Y/i ( X) \rightarrow C _ {1} $, etc. If $ ( X, i, Y) $ is a cofibration of pointed spaces, then for any pointed polyhedron $ K $ the induced sequence

$$ [ X, K] \leftarrow [ Y, K] \leftarrow [ Y/i ( X), K] \leftarrow [ C _ {1} , K] \leftarrow \dots $$

is an exact sequence of pointed sets; all terms of this sequence, from the fourth onward, are groups, and from the seventh onward — Abelian groups.

References

[1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)

Comments

In Western literature a cofibration always means what is here called a Borsuk pair.

How to Cite This Entry:
Cofibration. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cofibration&oldid=13585
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article